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A theory is presented for the effect of a tuned damper on a cantilever beam. The tuned damper consists of a mass connected through a link of viscoelastic material, with high loss factor, to a cantilever beam. Effective loss factors for the beam damper system are defined and experiments are described which confirm the main conclusions of the theory. Practical, light weight, damper unit designs which were used in the experimental investigation are also illustrated.
Many engineering problems can be solved using a linear approximation. In the Finite Element Analysis (FEA) the set of equations, describing the structural behaviour is then linear K d = F (1.1) In this matrix equation, K is the stiffness matrix of the structure, d is the nodal displacements vector and F is the external nodal force vector. Characteristics of linear problems is that the displacements are proportional to the loads, the stiffness of the structure is independent on the value of the load level. Though behaviour of real structures is nonlinear, e.g. displacements are not proportional to the loads; nonlinearities are usually unimportant and may be neglected in most practical problems.
Abstract: In the last few decades there has been a significant increase in the design strength and performance of different building materials. In particular, new methods, materials and admixtures for the production of concrete have allowed for strengths as high as 100 MPa to be readily available. In addition, the standard manufactured yield strength of reinforcing steel in Australia has increased from 400 MPa to 500 MPa. -- A perceived design advantage of higher-strength materials is that structural elements can have longer spans and be more slender than previously possible. An emerging problem with slender concrete members is that they can be more vulnerable to loading induced vibration. The damping capacity is an inherent fundamental quantity of all structural concrete members that affects their vibrational response. It is defined as the rate at which a structural member can dissipate the vibrational energy imparted to it. -- Generally damping capacity measurements, to indicate the integrity of structural members, are taken once the structure is in service. This type of non-destructive testing has been the subject of much research. The published non-destructive testing research on damping capacity is conflicting and a unified method to describe the effect of damage on damping capacity has not yet been proposed. -- Significantly, there is not one method in the published literature or national design codes, including the Australian Standard AS 3600-2001, available to predict the damping capacity of concrete beam members at the design stage. Further, little research has implemented full-scale testing with a view to developing damping capacity design equations, which is the primary focus of this thesis. -- To examine the full-range damping behaviour of concrete beams, two categories of testing were proposed. The categories are the 'untested' and 'tested' beam states. These beam states have not been separately investigated in previous work and are considered a major shortcoming of previous research on the damping behaviour of concrete beams. -- An extensive experimental programme was undertaken to obtain residual deflection and damping capacity data for thirty-one reinforced and ten prestressed concrete beams. The concrete beams had compressive strengths ranging between 23.1 MPa and 90.7 MPa, reinforcement with yield strengths of 400 MPa or 500 MPa, and tensile reinforcement ratios between 0.76% and 2.90%. The full- and half-scale beams tested had lengths of 6.0 m and 2.4 m, respectively. The testing regime consisted of a series of on-off load increments, increasing until failure, designed to induce residual deflections with increasing amounts of internal damage at which damping capacity (logarithmic decrement) was measured. -- The inconsistencies that were found between the experimental damping capacity of the beams and previous research prompted an initial investigation into the data obtained. It was found that the discrepancies were due to the various interpretations of the method used to extract damping capacity from the free-vibration decay curve. Therefore, a logarithmic decrement calculation method was proposed to ensure consistency and accuracy of the extracted damping capacity data to be used in the subsequent analytical research phase. -- The experimental test data confirmed that the 'untested' damping capacity of reinforced concrete beams is dependent upon the beam reinforcement ratio and distribution. This quantity was termed the total longitudinal reinforcement distribution. For the prestressed concrete beams, the 'untested' damping capacity was shown to be proportional to the product of the prestressing force and prestressing eccentricity. Separate 'untested' damping capacity equations for reinforced and prestressed concrete beams were developed to reflect these quantities. -- To account for the variation in damping capacity due to damage in 'tested' beams, a residual deflection mechanism was utilised. The proposed residual deflection mechanism estimates the magnitude of permanent deformation in the beam and attempts to overcome traditional difficulties in calculating the damping capacity during low loading levels. Residual deflection equations, based on the instantaneous deflection data for the current experimental programme, were proposed for both the reinforced and prestressed concrete beams, which in turn were utilised with the proposed 'untested' damping equation to calculate the total damping capacity. -- The proposed 'untested' damping, residual deflection and total damping capacity equations were compared to published test data and an additional series of test beams. These verification investigations have shown that the proposed equations are reliable and applicable for a range of beam designs, test setups, constituent materials and loading regimes.
Under steady state conditions, internal damping in prestressed concrete members may be less than 1% of critical if the initial prestress is sufficient to prevent tension cracks from developing. If tension cracks are allowed to develop, but on a miscroscopic scale, damping can be expected of the order of 2% of critical. If larger (visible) cracks are permitted to develop, higher damping would result. Under transient conditions, the amount of internal damping present in prestressed concrete members depends to a great extent on the past history of loading and on the amplitude of displacements produced. For those cases where members have been dynamically loaded only a few times to a given stress level which produces considerable cracking, damping can be expected anywhere in the range of 3 to 6% of critical. Magnitude and type of prestress in concrete members have an indirect influence on internal damping only because these parameters control the amount of cracking which can take place. (Author).
Abstract: Advances in construction materials and computational methods have made it possible to design and construct taller masts, buildings with increasingly slender frames, and bridges (and roof structures) with ever larger spans. In addition, masts, towers and new forms of construction such as offshore structures are being built in more hostile environments than previously contemplated. These evolving structures which keep extending the boundary of "normal" designs require that the designers take into account vibration of structures at the design stage to a much greater extent than they have done in the past. -- The slenderness of modern structures and the large magnitude of the loads that many of them must carry also make it imperative that such structures be designed for stresses induced by dynamic disturbances. The response of a structure to a dynamically applied load may be many times greater than its response to the same load applied statically. The relationship between a structure's static and dynamic responses depends primarily on its damping characteristics and on its natural periods of vibration. In fact, damping is one of the most significant contributors to the dynamic response of high-rise buildings, bridges, tall chimneys and other slender structures considered to be significantly affected by dynamic forces. -- Under a severe lateral dynamic loading condition, the structure that is likely to survive is one whose members are sufficiently ductile to absorb and dissipate energy by elastic and/or inelastic deformation. This requires the designer to realistically assess the possible levels of strength in flexural and shear elements. Thus, in designing such a concrete structure, it is important to understand and determine the ability of the structure to absorb energy under an external impulsive force. At this stage, information in this regard is lacking in published literature and the ability of the constituent elements of the structure to absorb energy is not well understood. This, for example, is true for reinforced and partially prestressed concrete beam, especially the cracked ones. In particular, no simple and accurate formulae are available to evaluate the damping ratios of reinforced and partially prestressed concrete beams cracked or otherwise, for use in the dynamic design of civil engineering structures. It is this area which forms the primary focus of this research. -- In this research, an extensive test programme has been carried out to study the cracking and damping behaviour of reinforced and partially prestressed concrete beams. The tests were carried out in two stages and involved a total of 30 reinforced and partially prestressed beams. Nine reinforced and 12 partially prestressed simply supported full-size box beams were tested at the first stage. Tested at the second stage were 2 simply supported and 3 two-equal span continuous reinforced full-size box beams and 4 solid rectangular full-size simply supported reinforced beams. For all the beams, at each level of loading, measurements were made of instantaneous and residual crack widths, instantaneous and residual concrete strains, and mid-span deflections. Each beam was also subjected to free vibration tests to measure its logarithmic decrement of damping corresponding to each load level. -- Based on the experimental results, two empirical formulae have been developed for predicting logarithmic decrement of damping separately in reinforced and partially prestressed concrete beams. These formulae predict damping from the residual crack widths of the beams. For these formulae to be of practical use, a formula relating the residual crack widths of concrete beams to the instantaneous average crack widths was developed. In addition, a unified formula was derived for the prediction of the instantaneous average crack widths based on the general beam parameters. As an alternative, separate formulae are also presented for predicting residual crack widths using mid-span deflections of reinforced and partially prestressed beams. These further enhance the practicability of the proposed damping formulae. -- In an effort to verify the accuracy and reliability of the proposed formulae, comparative studies are carried out based on the author's own laboratory test results as well as those available in published literature. In total, 104 full-size reinforced and prestressed concrete solid and box beams are involved in the comparative study. In general, good correlations are obtained for instantaneous and residual average crack widths and for logarithmic decrement of damping values. These are true for both reinforced and partially prestressed concrete beams.